具有任意非线性项的非局部非线性薛定谔方程精确解的稳定性
Stability of exact solutions of a nonlocal and nonlinear Schrödinger equation with arbitrary nonlinearity
摘要：
研究了具有任意非线性参数$κ$的$1$ + $1$维非局部非线性薛定谔方程组的孤波解.虽然我们在这里研究的系统是由Yang（Phys. Rev. E，98（2018），042202）首次报道的完全可积的情况$κ=1$，我们扩展了它的考虑，并提供了孤子稳定性和不稳定性的标准作为$κ$的函数。特别是，我们表明，$κ<2$的解决方案是稳定的，而$κ>2$，他们受到崩溃或爆破。在$κ=2$的临界点，存在一个爆炸或坍塌所必需的临界质量。此外，我们发现有一个简单的一个组件的非局部拉格朗日管理系统的动力学是服从集体坐标近似。为此，我们引入了一个试验波函数与两个集体坐标研究周围的精确解的小振荡。在集体坐标近似下，我们得到了宽度参数的小振荡频率的解析表达式。我们还讨论了四个集体坐标近似，这反过来又打破了对称性的精确解，允许平移运动。在后一种情况下发现的随后的振荡捕获了孤子对小平移的响应。最后，我们的结果进行了比较与系统的数值模拟。
英文摘要：
This work focuses on the study of solitary wave solutions to a nonlocal, nonlinear Schrödinger system in $1$+$1$ dimensions with arbitrary nonlinearity parameter $κ$. Although the system we study here was first reported by Yang (Phys. Rev. E, 98 (2018), 042202) for the fully integrable case $κ=1$, we extend its considerations and offer criteria for soliton stability and instability as a function of $κ$. In particular, we show that for $κ<2$ the solutions are stable whereas for $κ>2$ they are subject to collapse or blowup. At the critical point of $κ=2$, there is a critical mass necessary for blowup or collapse. Furthermore, we show there is a simple one-component nonlocal Lagrangian governing the dynamics of the system which is amenable to a collective coordinate approximation. To that end, we introduce a trial wave function with two collective coordinates to study the small oscillations around the exact solution. We obtain analytical expressions for the small oscillation frequency for the width parameter in the collective coordinate approximation. We also discuss a four collective coordinate approximation which in turn breaks the symmetry of the exact solution by allowing for translational motion. The ensuing oscillations found in the latter case capture the response of the soliton to a small translation. Finally, our results are compared with numerical simulations of the system.
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